Journal of Convex Analysis 14 (2007), No. 2, 433--454
Copyright Heldermann Verlag 2007
Infinite Dimensional Clarke Generalized Jacobian
Zsolt Páles
Institute of Mathematics, University of Debrecen, 4010 Debrecen Pf. 12, Hungary
pales@math.klte.hu
Vera Zeidan
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A.
zeidan@math.msu.edu
We extend for a locally Lipschitz function the notion of Clarke's generalized Jacobian to the setting where the domain lies in an infinite dimensional normed space. When the function is real-valued this notion reduces to the Clarke's generalized gradient. Using this extension, we obtain an exact smooth-nonsmooth chain rule from which the sum rule and the product rule follow. Also an exact formula for the generalized Jacobian of piecewise differentiable functions will be provided.
Keywords: Generalized Jacobian, chain rule, sum rule, piecewise smooth functions.
MSC: 49A52, 58C20
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